Solve 6(2-x)9/2x=2(-3x+6)/4

$$6 ( 2 - x ) \frac { 9 } { 2 } x = \frac { 2 ( - 3 x + 6 ) } { 4 }$$

Solve for x

$$x=2$$

$$x=\frac{1}{18}\approx 0.055555556$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$24\left(2-x\right)\times \frac{9}{2}x=2\left(-3x+6\right)$$

Express 24\times \frac{9}{2} as a single fraction.

$$\frac{24\times 9}{2}\left(2-x\right)x=2\left(-3x+6\right)$$

Multiply 24 and 9 to get 216.

$$\frac{216}{2}\left(2-x\right)x=2\left(-3x+6\right)$$

Divide 216 by 2 to get 108.

$$108\left(2-x\right)x=2\left(-3x+6\right)$$

Use the distributive property to multiply 108 by 2-x.

$$\left(216-108x\right)x=2\left(-3x+6\right)$$

Use the distributive property to multiply 216-108x by x.

$$216x-108x^{2}=2\left(-3x+6\right)$$

Use the distributive property to multiply 2 by -3x+6.

$$216x-108x^{2}=-6x+12$$

Add 6x to both sides.

$$216x-108x^{2}+6x=12$$

Combine 216x and 6x to get 222x.

$$222x-108x^{2}=12$$

Subtract 12 from both sides.

$$222x-108x^{2}-12=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-108x^{2}+222x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -108 for a, 222 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-222±\sqrt{222^{2}-4\left(-108\right)\left(-12\right)}}{2\left(-108\right)}$$

Square 222.

$$x=\frac{-222±\sqrt{49284-4\left(-108\right)\left(-12\right)}}{2\left(-108\right)}$$

Multiply -4 times -108.

$$x=\frac{-222±\sqrt{49284+432\left(-12\right)}}{2\left(-108\right)}$$

Multiply 432 times -12.

$$x=\frac{-222±\sqrt{49284-5184}}{2\left(-108\right)}$$

Add 49284 to -5184.

$$x=\frac{-222±\sqrt{44100}}{2\left(-108\right)}$$

Take the square root of 44100.

$$x=\frac{-222±210}{2\left(-108\right)}$$

Multiply 2 times -108.

$$x=\frac{-222±210}{-216}$$

Now solve the equation x=\frac{-222±210}{-216} when ± is plus. Add -222 to 210.

$$x=-\frac{12}{-216}$$

Reduce the fraction \frac{-12}{-216} to lowest terms by extracting and canceling out 12.

$$x=\frac{1}{18}$$

Now solve the equation x=\frac{-222±210}{-216} when ± is minus. Subtract 210 from -222.

$$x=-\frac{432}{-216}$$

Divide -432 by -216.

$$x=2$$

The equation is now solved.

$$x=\frac{1}{18} x=2$$

Steps for Completing the Square

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$24\left(2-x\right)\times \frac{9}{2}x=2\left(-3x+6\right)$$

Express 24\times \frac{9}{2} as a single fraction.

$$\frac{24\times 9}{2}\left(2-x\right)x=2\left(-3x+6\right)$$

Multiply 24 and 9 to get 216.

$$\frac{216}{2}\left(2-x\right)x=2\left(-3x+6\right)$$

Divide 216 by 2 to get 108.

$$108\left(2-x\right)x=2\left(-3x+6\right)$$

Use the distributive property to multiply 108 by 2-x.

$$\left(216-108x\right)x=2\left(-3x+6\right)$$

Use the distributive property to multiply 216-108x by x.

$$216x-108x^{2}=2\left(-3x+6\right)$$

Use the distributive property to multiply 2 by -3x+6.

$$216x-108x^{2}=-6x+12$$

Add 6x to both sides.

$$216x-108x^{2}+6x=12$$

Combine 216x and 6x to get 222x.

$$222x-108x^{2}=12$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-108x^{2}+222x=12$$

Divide both sides by -108.

$$\frac{-108x^{2}+222x}{-108}=\frac{12}{-108}$$

Dividing by -108 undoes the multiplication by -108.

$$x^{2}+\frac{222}{-108}x=\frac{12}{-108}$$

Reduce the fraction \frac{222}{-108} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{37}{18}x=\frac{12}{-108}$$

Reduce the fraction \frac{12}{-108} to lowest terms by extracting and canceling out 12.

$$x^{2}-\frac{37}{18}x=-\frac{1}{9}$$

Divide -\frac{37}{18}, the coefficient of the x term, by 2 to get -\frac{37}{36}. Then add the square of -\frac{37}{36} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{37}{18}x+\left(-\frac{37}{36}\right)^{2}=-\frac{1}{9}+\left(-\frac{37}{36}\right)^{2}$$

Square -\frac{37}{36} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{37}{18}x+\frac{1369}{1296}=-\frac{1}{9}+\frac{1369}{1296}$$

Add -\frac{1}{9} to \frac{1369}{1296} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{37}{18}x+\frac{1369}{1296}=\frac{1225}{1296}$$

Factor x^{2}-\frac{37}{18}x+\frac{1369}{1296}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{37}{36}\right)^{2}=\frac{1225}{1296}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37}{36}\right)^{2}}=\sqrt{\frac{1225}{1296}}$$

Simplify.

$$x-\frac{37}{36}=\frac{35}{36} x-\frac{37}{36}=-\frac{35}{36}$$

Add \frac{37}{36} to both sides of the equation.

$$x=2 x=\frac{1}{18}$$